When interfering objects occlude a scene, the visual system restores the occluded information. Similarly, when a sound of interest (a "foreground" sound) is interrupted (occluded) by loud noise, the auditory system restores the occluded information. This process, called auditory induction, can be exploited to create a continuity illusion. When a segment of a foreground sound is deleted and loud noise fills the missing portion, listeners incorrectly report hearing the foreground continuing through the noise. Here we reveal the neurophysiological underpinnings of illusory continuity in single-neuron responses from awake macaque monkeys' primary auditory cortex (A1). A1 neurons represented the missing segment of occluded tonal foregrounds by responding to discontinuous foregrounds interrupted by intense noise as if they were responding to the complete foregrounds. By comparison, simulated peripheral responses represented only the noise and not the occluded foreground. The results reveal that many A1 single-neuron responses closely follow the illusory percept.